Control of systems with multiplicative observation noise

We consider the control of a linear system observed over multiplicative-noise. Specifically, the controller must stabilize the system using a control action based on observations of the system state that have been multiplied by i.i.d. random variables. While there is a long history of work on this fundamental problem, much of it has focused on understanding the performance of linear controllers, and the optimal control strategy for such a system remains unknown. In this paper, we consider the case of uniform multiplicative observation noise, and provide a non-linear control strategy based on the maximum a-posteriori (MAP) estimator of the state. We explicitly compute the convergence rates of different moments of the system under this control strategy, and find that the MAP-based strategy outperforms the best memoryless linear strategy when the "signal-to-noise" ratio (SNR) of the multiplicative noise, i.e. the ratio of the mean to the standard deviation, is low. In the high SNR regime we see that the MAP strategy is also a linear memoryless strategy, however, it is suboptimal and is outperformed by the optimal linear controller.